Introduction to Natural Computation
Lecture 18
Random Boolean Networks
Alberto Moraglio
1 / 25 Random Boolean Networks
Extension of Cellular Automata.
Introduced by Stuart Kauffman (1969) as model for genetic regulatory networks.
Uncover fundamental principles of living systems.
Hypothesis: living organisms can be constructed from random elements.
Simplification: Boolean states for all nodes in the network.
2 / 25 Definition
Random Boolean Network (RBN) A random Boolean network consists of N nodes, each with a state 0 or 1 K edges for each node to K different randomly selected nodes a lookup table for each node that determines the next state, according to states of its K neighbours.
Last few lectures: fixed graph, random process This time: random graph, deterministic process Note: randomness used to create network, afterwards network remains fixed!
3 / 25 Relation to Cellular Automata
http://www.metafysica.nl/boolean.html
4 / 25 Lookup Tables
Recall rules from Cellular Automata:
RBN: each node has its own random lookup table: inputs at time t state at time t + 1 000 1 001 0 010 0 011 1 100 1 101 0 110 1 111 1
5 / 25 Example of a Simple RBN
Gershenson C. Introduction to Random Boolean Networks. ArXiv Nonlinear Sciences e-prints. 2004;
6 / 25 States
State space has size 2N .
After at most 2N + 1 steps a state will be repeated.
7 / 25 States
State space has size 2N .
After at most 2N + 1 steps a state will be repeated.
Attractors As the system is deterministic, it will get stuck in an attractor. one state → point attractor / steady state two or more states → cycle attractor / state cycle The set of states that flow towards an attractor is called attractor basin.
8 / 25 Number of RBNs
Q: How many possible functions exists for each node?
9 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
10 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
Q: How many possible wirings to K neighbours exist for each node?
11 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations.
12 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations.
Q: How large is the number of possible networks for given N, K?
13 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations.
Q: How large is the number of possible networks for given N, K? A: Pretty big!
14 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations.
Q: How large is the number of possible networks for given N, K? A: Pretty big! K N 22 N! (N − K)! !
15 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations.
Q: How large is the number of possible networks for given N, K? A: Pretty big! K N 22 N! (N − K)! !
N = 3,K = 3 : 3623878656
16 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations.
Q: How large is the number of possible networks for given N, K? A: Pretty big! K N 22 N! (N − K)! !
N = 3,K = 3 : 3623878656 N = 4,K = 3 : 1424967069597696
17 / 25 Number of RBNs
Q: How many possible functions exists for each node? K A: All possible 0-1-choices for 2K inputs: 22 .
Q: How many possible wirings to K neighbours exist for each node? A: There are N!/(N − K)! ordered combinations.
Q: How large is the number of possible networks for given N, K? A: Pretty big! K N 22 N! (N − K)! !
N = 3,K = 3 : 3623878656 N = 4,K = 3 : 1424967069597696 N = 8,K = 4 : 21593035501811706443110335226483118854343255930579818905600000000
18 / 25 Order vs. Chaos
Random Boolean networks can be in three phases: ordered, chaotic, and critical.
Gershenson C. Introduction to Random Boolean Networks. ArXiv Nonlinear Sciences e-prints. 2004;
19 / 25 Order vs. Chaos
Ordered regime Occurs when K ≤ 2. System is insensitive to initial conditions and “mutations”: flip the state of a node change a connection change the lookup table of a node “Damage” typically does not spread far.
20 / 25 Order vs. Chaos
Ordered regime Occurs when K ≤ 2. System is insensitive to initial conditions and “mutations”: flip the state of a node change a connection change the lookup table of a node “Damage” typically does not spread far.
Chaotic regime Occurs when K ≥ 3.
Small changes can have a huge impact–“butterfly effect”.
Similar states tend to diverge.
21 / 25 View of an Attractor Basin
N = 13,K = 3
http://www.metafysica.nl/boolean.html
22 / 25 All Attractor Basins
N = 13,K = 3
http://www.metafysica.nl/boolean.html from Andrew Wuensche’s DLLab gallery
23 / 25 Other Update Schemes
Attractor cycles occur because the system is deterministic and all nodes are updated synchronously.
Criticism about synchronicity: genes do not march in step!
Variants of RBN Asynchronous RBNs: randomly choose a node to be updated. Deterministic asynchronous RBNs: create random periods Pi,Qi for each node that remain fixed afterwards. Update node i at time t if t mod Pi ≡ Qi.
Other update schemes can change the behaviour significantly.
24 / 25 Further Reading
Gershenson C. Introduction to Random Boolean Networks. ArXiv Nonlinear Sciences e-prints. 2004;.
25 / 25